A CENTRAL LIMIT THEOREM FOR STRONG NEAR-EPOCH DEPENDENT RANDOM VARIABLES

In this paper, a central limit theorem for strong near-epoch dependent sequences of random variables introduced in [9] is showed. Under the same moments condition, the authors essentially weaken the "size" requirement mentioned in other papers about near epoch dependence.     

The functional central limit theorem for linear processes with strong near-epoch dependent innovations

This paper discusses linear processes with innovations exhibiting asymptotic weak dependence by being strong near-epoch dependent functions of mixing processes. The functional central limit theorem for the normalized partial sum process is established. The conditions given essentially improve on existing results in the literature in terms of the “size” requirement for the amount of dependence. It is also shown that two important econometric models, ARMA and GARCH models, are strong near-epoch dependent sequences.     

The invariance principle for fractionally integrated processes with strong near-epoch dependent innovations

In this paper, we show the invariance principle for the partial sum processes of fractionally integrated processes, otherwise known as I(d + m) processes, where |d| < 1/2 and m is a nonnegative integer, with strong near-epoch dependent innovations. The results are applied to the test of unit root. The conditions given improve previous results in the literature concerning fractionally integrated processes.     

Cesàro α-Integrability and Laws of Large Numbers-II

For a sequence of random variables, a new set of properties called Cesàro α-Integrability and Strong Cesàro α-Integrability was recently introduced in an earlier paper and these properties were used to prove several new laws of large numbers, namely both Strong and Weak Laws of Large Numbers for pairwise-independent random variables as well as WLLN for some dependent sequences of random variables. In this paper, a set of weaker conditions called Residual Cesàro α-Integrability and Strong Residual Cesàro α-Integrability are introduced and significant improvements over earlier results are obtained. In addition, new results on L _p -convergence, for 0 < p < 2, and SLLN for some dependent sequences are proved.      

Lp-Approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables

The properties of L₂-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are “two-wing” averages of martingale differences. The results constitute the first significant advancement in the theory of L₂-approximable sequences since 1976 when Moussatat introduced a narrower notion of L₂-generated sequences. The method relies on a study of certain linear operators in the spaces L_p and l_p. A criterion of L_p-approximability is given. The results are new even when the weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.     

Computable examples of the maximal Lyapunov exponent

Summary Some new examples are given of sequences of matrix valued random variables for which it is possible to compute the maximal Lyapunov exponent. The examples are constructed by using a sequence of stopping times to group the original sequence into commuting blocks. If the original sequence is the outcome of independent Bernoulli trials with success probability p, then the maximal Lyapunov exponent may be expressed in terms of power series in p, with explicit formulae for the coefficients. The convexity of the maximal Lyapunov exponent as a function of p is discussed, as is an application to branching processes in a random environment.     

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??Examining the conditions of positively or negatively associated sequences of random variables obeying the strong law of large numbers provided by Alexander, the sequences of Gaussian random variables, nonnegative and uniformly bounded sequences of random variables with general dependent structure were studied, and the sufficient conditions for they obeying the strong law of large numbers were given. At last, an example for Gaussian sequence satisfying the strong law of large numbers was given.     

Measuring monotony in two-dimensional samples

This note introduces a monotony coefficient as a new measure of the monotone dependence in a two-dimensional sample. Some properties of this measure are derived. In particular, it is shown that the absolute value of the monotony coefficient for a two-dimensional sample is between |r| and 1, where r is the Pearson's correlation coefficient for the sample; that the monotony coefficient equals 1 for any monotone increasing sample and equals ?1 for any monotone decreasing sample. This article contains a few examples demonstrating that the monotony coefficient is a more accurate measure of the degree of monotone dependence for a non-linear relationship than the Pearson's, Spearman's and Kendall's correlation coefficients. The monotony coefficient is a tool that can be applied to samples in order to find dependencies between random variables; it is especially useful in finding couples of dependent variables in a big dataset of many variables. Undergraduate students in mathematics and science would benefit from learning and applying this measure of monotone dependence.     

On the conditional expectation with respect to a sequence of independent σ-fields

Summary In the paper we characterize those sequences of random variables which are conditional expectations of a p-integrable random variable with respect to a given sequence of independent -fields.     

On Lower Bounds for Lp Norms in the Central Limit Theorem for Independent and m-Dependent Random Variables

We derive lower bounds for L_p norms , in the central limit theorem for independent and m–dependent random variables with finite fifth order absolute moments and for independent and m–dependent identically distributed random variables with fourth order moments.     

Mean Convergence Theorems for Weighted Sums of Arrays of Residually <Emphasis Type="Italic">h</Emphasis>-integrable Random Variables Concerning the Weights under Dependence Assumptions

From the classical notion of uniform integrability of a sequence of random variables, a new concept called residual h-integrability is introduced for an array of random variables, concerning an array of constants, which is weaker than other previous related notions of integrability. Martingale difference, pairwise negative quadrant dependence, tail φ-mixing property and L _p -mixingale are four special kinds of dependence structures, where 1≤p≤2. By relating the residual h-integrability with such these dependence assumptions, some conditions are formulated under which mean convergence theorems for weighted sums of arrays of random variables are established, and many earlier results are explained as the special cases of the ones appearing in our present work.      

On the strong law of large numbers for sequences of random variables without the independence condition

New sufficient conditions for the applicability of the strong law of large numbers to a sequence of dependent random variables X ₁, X ₂, …, with finite variances are established. No particular type of dependence between the random variables in the sequence is assumed. The statement of the theorem involves the classical condition Σ _n ^∞ (log₂ n)²/n ² < ∞, which appears in various theorems on the strong law of large numbers for sequences of random variables without the independence condition.     

Quasi-random sequences by power residues

For any primep, quasi-random sequences are constructed whose elements progressively divide the unit interval intop,p ²,p ³, ... equal parts. This property of uniformity is obtained by a slight modification of the multiplicative congruential scheme for the generation of random numbers. The extension to the unit square is also discussed and comparison with other quasi-random and random sequences is made through examples.     

p-Norm bounds on the expectation of the maximum of a possibly dependent sample

Let X₁, X₂,…, X_n be identically distributed possibly dependent random variables with finite pth absolute moment assumed without loss of generality to be equal to 1. Denote the order statistics by X_1:n, X_2:n,…, X_n:n. Bounds are derived for E(X_n:n) when it is assumed that the X_i's are (i) arbitrarily dependent and (ii) independent. The effect of assuming a symmetric common distribution for the X_i's is discussed. Analogous bounds are described for the expected range of the sample. Bounds on expectations of general linear combinations of order statistics are described in the independent case.     

On hazard rate ordering of the sums of heterogeneous geometric random variables

In this paper, we treat convolutions of heterogeneous geometric random variables with respect to the p-larger order and the hazard rate order. It is shown that the p-larger order between two parameter vectors implies the hazard rate order between convolutions of two heterogeneous geometric sequences. Specially in the two-dimensional case, we present an equivalent characterization. The case when one convolution involves identically distributed variables is discussed, and we reveal the link between the hazard rate order of convolutions and the geometric mean of parameters. Finally, we drive the “best negative binomial bounds” for the hazard rate function of any convolution of geometric sequence under this setup.     

Limit theorems for sums of dependent random variables

Summary In Lai and Stout [7] the upper half of the law of the iterated logarithm (LIL) is established for sums of strongly dependent stationary Gaussian random variables. Herein, the upper half of the LIL is established for strongly dependent random variables {X _i} which are however not necessarily Gaussian. Applications are made to multiplicative random variables and to f(Z _i ) where the Z _i are strongly dependent Gaussian. A maximal inequality and a Marcinkiewicz-Zygmund type strong law are established for sums of strongly dependent random variables X _i satisfying a moment condition of the form E¦S _a,n ¦^pg(n), where , generalizing the work of Serfling [13, 14].Research supported by the National Science Foundation under grant NSF-MCS-78-09179Research supported by the National Science Foundation under grant NSF-MCS-78-04014     

On the rate of approximation in the central limit theorem for dependent random variables and random vectors

Large “O” and small “o” approximations of the expected value of a class of smooth functions (f C^r(R)) of the normalized partial sums of dependent random variable by the expectation of the corresponding functions of normal random variables have been established. The same types of approximations are also obtained for dependent random vectors. The technique used is the Lindberg-Levy method generalized by Dvoretzky to dependent random variables.     

Stochastic Comparisons and Dependence among Concomitants of Order Statistics

Let (X_i, Y_i) i=1, 2, …, n be n independent and identically distributed random variables from some continuous bivariate distribution. If X_(r) denotes the rth ordered X-variate then the Y-variate, Y_[r], paired with X_(r) is called the concomitant of the rth order statistic. In this paper we obtain new general results on stochastic comparisons and dependence among concomitants of order statistics under different types of dependence between the parent random variables X and Y. The results obtained apply to any distribution with monotone dependence between X and Y. In particular, when X and Y are likelihood ratio dependent, it is shown that the successive concomitants of order statistics are increasing according to likelihood ratio ordering and they are TP₂ dependent in pairs. If we assume that the conditional hazard rate of Y given X=x is decreasing in x, then the concomitants are increasing according to hazard rate ordering and are dependent according to the right corner set increasing property. Finally, it is proved that if Y is stochastically increasing in X, then the concomitants of order statistics are stochastically increasing and are associated. Analogous results are obtained when the variables X and Y are negatively dependent. We also prove that if the hazard rate of the conditional distribution of Y given X=x is decreasing in x and y, then the concomitants have DFR (decreasing failure rate) distributions and are ordered according to dispersive ordering.     

Central Limit Theorems for Asymptotically Negatively Associated Random Fields

Abstract The aim of this paper is to investigate the central limit theorems for asymptotically negatively dependent random fields under lower moment conditions or the Lindeberg condition. Results obtained improve a central limit theorem of Roussas [11] for negatively assiated fields and the main results of Su and Chi [18], and also include a central limit of theorem for weakly negatively associated random variables similar to that of Burton et al. [20]. Research supported by National Natural Science Foundation of China (No. 19701011)     

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.